A Realizable Constitutive Model for Fiber-Reinforced Neo-Hookean Solids

Abstract

We provide a closed-form analytical expression for the effective stored-energy function of Neo-Hookean solids reinforced by a random distribution of anisotropic cylindrical fibers, subject to general finite-strain loading conditions. The expression is obtained by means of a homogenization constitutive theory recently proposed by the authors (Lopez-Pamies O., Idiart M.I. Fiber-reinforced hyperelastic solids: A realizable homogenization constitutive theory. J. Eng. Math., submitted) to determine the mechanical response of fiber-reinforced hyperelastic solids. The central idea in this theory is to devise a special class of random microgeometries —by means of an iterated homogenization procedure together with an exact dilute result for sequential laminates— that allows to compute exactly the macroscopic response of the resulting fiber-reinforced solid. The derived constitutive relations incorporate direct microstructural information up to the two-point statistics. Since the resulting effective stored-energy function is realizable, in the sense that it is exact for a given class of microgeometries, it is guaranteed to be theoretically sound and to give physically sensible predictions. The predictions of the model are illustrated through stress-strain relations and loss of ellipticity criteria. The mechanical stability of fiber-reinforced Neo-Hookean solids is analyzed in the light of the new predictions.